The Fourier Transforms

The Fourier transform was named after Joseph Fourier, a French mathematician. A Fourier Transform takes a function to its frequency components.

Properties of a Fourier Transform:

Properties of a Fourier Transform:

Linearity

   ${\mathcal {F}}[a*x(t)+b*y(t)]=a*X(f)+b*Y(f)$

= Shifting the function changes the phase of the spectrum

   Failed to parse (SVG with PNG fallback (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \mathcal{F}[x(t-a)] = X(t)e^{-j2\pi f a}\}

Frequency and amplitude are affected when changing spatial scale inversely

   ${\mathcal {F}}[x(a*t)]={\frac {1}{a}}X({\frac {f}{a}})$

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